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Recent experiments have discovered multipolar orders in a variety of$d$-orbital Mott insulators. Motivated by uncovering the exchange interactionswhich underlie octupolar order proposed in the osmate double perovskites, westudy a two-site model using exact diagonalization on a five-orbitalHamiltonian, incorporating spin-orbit coupling (SOC) and interactions, andincluding both intra-orbital and inter-orbital hopping. Using an exactSchrieffer-Wolff transformation, we then extract an effective pseudospinHamiltonian for the non-Kramers doublets, uncovering dominant ferrooctupolarcoupling driven by the interplay of two distinct intra-orbital hopping terms.Using classical Monte Carlo simulations on the face-centered cubic lattice, weobtain a ferrooctupolar transition temperature which is in good agreement withexperiments on the osmate double perovskites. We also explore the impact ofuniaxial strain and dimensional tuning via ultrathin films, which are shown toinduce a transverse field on the Ising octupolar order. This suppresses $T_c$and potentially allows one to access octupolar Ising quantum critical points.We discuss possible implications of our results for a broader class ofmaterials which may host such non-Kramers doublet ions.

Octupolar order and Ising quantum criticality tuned by strain and dimensionality: Application to $d$-orbital Mott insulators